Solution to Problem 83-4*: A covering problem

Solution to Problem 83-4*: A covering problem

Citation for published version (APA):

Lossers, O. P. (1984). Solution to Problem 83-4*: A covering problem. SIAM Review, 26(1), 124-. https://doi.org/10.1137/1026013

DOI:

10.1137/1026013

Document status and date: Published: 01/01/1984 Document Version:

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124 PROBLEMS AND SOLUTIONS

Incidentally,thisproblemiscloselyrelatedtotheevaluationof

n n

S--n=

(j) (nk)max(j,k),

S_n

_..k

(j)

(nk)min(j,k).

Indeed, since

S---,

_{_Sn}

_2Sn

and

S---,

₊

_{_S}

_{(7)(7,)(J +}

k) n.

2,

we have

-n

n.

22n-l

+ Sand_S-

n.

22-1

_Sn.

Editorialnote.

Can

oneexplicitlysumtheanalogousextensions of

_Sn

and

_Sn

toruth ordersummations?

[M.S.K.]

Also solvedby J. C. BUTCHER (University ofAuckland,Auckland, New Zealand), C. GEORHIOU (University of Patras,

Patras,

Greece), H. VAN HAERINGEN (Delft

University ofTechnology, Delft, theNetherlands), S. D. HENDRV (Baltimore,MD), M.

HOFFMAN (Memorial University ofNewfoundland), E. Hou

(no

affiliation or address given), A. A..lAGERS (Technische Hogeschool Twente, Enschede, the Netherlands), M. S..IANKOVlC

(PHH

Engineering, Calgary, Alberta), R. A. JOHNS, (E-Systems,

Garland, TX), W. B. JORDAN (Scotia,

NY),

I. KINNMARI< (Princeton, N‘l), O. P.

LOSSERS (Eindhoven University ofTechnology, Eindhoven, theNetherlands), W. A. J. LUXEMBURG (California Institute ofTechnology), ‘l. PIEPERS (Castricum, the Nether-lands), M. RENARDV (University of Wisconsin), ‘l. RtJPPERT (Wirtschafts Universitit Wien, Wien, Augasse), O. G. RUEHR (Michigan Technological University), A. SIDI (NASA, Cleveland, OH), .l.A. WILSON

(Iowa

State University), P. Y. Wu (National

Chiao

Tung

University, Hsinchu,Taiwan)andthe proposer.

ACoveringProblem

Problem83-4*,byJ. VIJAY(University ofFlorida).

Let Mbe a set ofm fixed anddistinct points in the plane. Define a hoopofradius

r >0 to beanymaximalsubset ofMthatissimultaneously coverablebyacircleofradius

r. Obviously, the number of such hoops depends on both r and the locations ofthe m

points.Forexample,ifrisverysmall then eachpointis ahoop,and ifrisverylarge then

theonly hoopisMitself.

It is conjectured that for a given m, the number of hoops cannot exceed 3m

irrespective of thelocationsof thempoints and the value ofr.Thisboundis attained inthe

limitwhenm-- and the pointsarein anequilateral triangularlattice with2r being the

distancebetween any point and any ofits sixclosest neighbors.

Solution by O. P. LOSSERS (Eindhoven University of Technology, Eindhoven, the

Netherlands).

Theconjectureisfalse!

Consider inthe equilateral triangularlatticethe following configuration.

T

Itis a matterof straightforward checking that everyhoophas the above form(apartfrom

arotation overamultiple of

r/3).

Sinceeveryhoop hasaunique

"top"

Tand any pointis